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(CNN) -- Citing sources close to the kidnappers, Alrai Television in Kuwait reported Friday that U.S. journalist Jill Carroll's kidnappers set "a final deadline of February 26" for their demands to be met.

It would be the second deadline set by her abductors, who have repeatedly said they will kill Carroll if the United States does not release all female prisoners it has in custody in Iraq.

The kidnappers said they would be adhering to "rightful law" in killing Carroll, according to the private Kuwaiti station's chairman, Jassim Boodai.

The kidnappers said that Carroll is being held at a "safe house" in central Baghdad owned by one of the abductors and lives with a group of women with whom she is "sharing the house chores," Boodai said.

"She is in good health," he added.

In a video that appeared Thursday on the same station, Carroll said she was doing well, but urged the United States to meet her captors' demands quickly.

"I'm here. I'm fine. Please, just do whatever they want. Give them whatever they want as quickly as possible. There is very short time. Please do it fast," she said. (Watch Carroll's plea -- :40)

Carroll, who wore a hijab, or Muslim headdress, during the video also said that it was February 2. She appeared more composed than she had during a video broadcast January 30, in which she was weeping.

"I sent you a letter written by my hand that you wanted more evidence, so we're sending you this new letter now just to prove that I am with the mujahedeen," Carroll says in English on the tape.

Alrai -- which means "The Opinion" -- said Thursday it had a copy of the letter Carroll referred to and was planning to give it to authorities. The video and the letter were dropped off at the station's Baghdad office, Alrai said.

Boodai said that Friday's information came from the same sources as Thursday's tape and letter, which contains "personal information" and was handwritten by Carroll.

"Our concern is the safety of Jill Carroll, and that's why we are fully cooperating with authorities," Boodai said.

The 28-year-old freelance writer for The Christian Science Monitor was kidnapped January 7 in western Baghdad. Her Iraqi interpreter was killed, but her Iraqi driver escaped.

Boodai said Friday that the kidnappers deny having any role in the death of Carroll's translator.

Carroll, who has been reporting from the Middle East for three years, was planning to meet with Iraqi politician Adnan al-Dulaimi for an interview, but he was not there, according to The Christian Science Monitor, which interviewed her driver.

As the three attempted to drive off, their vehicle was stopped by the insurgents, the paper reported. (Full story)

Carroll's family released a statement Thursday, saying, "The family is hopeful and grateful to all those working on Jill's behalf."

The United States has released five Iraqi women prisoners since Carroll's kidnapping, but Washington said those releases had nothing to do with the kidnappers' demands. Four others are still in custody.

The most recent tape is the third on which Carroll has appeared. The first two were broadcast on the Arabic-language network Al-Jazeera. The second tape, released January 30, bore the logo of a group called Brigades of Vengeance, which has claimed responsibility for her kidnapping.